Difference between revisions of "Direct lattice"
From Online Dictionary of Crystallography
m (links) |
BrianMcMahon (talk | contribs) m (Tidied translations.) |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | < | + | <font color="blue">Réseau direct</font> (''Fr''). <font color="red">Direktes Gitter</font> (''Ge''). <font color="black">Reticolo diretto</font> (''It''). <font color="purple">直格子</Font> (''Ja''). <font color="green">Red directa</font> (''Sp''). |
== Definition == | == Definition == | ||
| Line 6: | Line 6: | ||
'''t<sub>2</sub>''' and '''t<sub>3</sub>'''. One distinguishes two kinds of lattices, the ''vector lattices'' and the ''point lattices''. | '''t<sub>2</sub>''' and '''t<sub>3</sub>'''. One distinguishes two kinds of lattices, the ''vector lattices'' and the ''point lattices''. | ||
| − | Any translation '''t''' = ''u<sup>i</sup>'' '''t<sub>i</sub>''' (''u<sup>i</sup>'' arbitrary integers) is also a translation of the pattern and the infinite set of all translation vectors of a crystal pattern is the '''vector lattice''' '''L''' of this crystal pattern. | + | Any translation '''t''' = ''u<sup>i</sup>'' '''t<sub>i</sub>''' (''u<sup>i</sup>'' arbitrary integers) is also a translation of the pattern and the infinite set of all translation vectors of a crystal pattern is the '''vector lattice''' '''L''' of this [[crystal pattern]]. |
Given an arbitrary point ''P'' in point space, the set of all the points ''P<sub>i</sub>'' deduced from one of them by a translation '''PP<sub>i</sub>''' = '''t<sub>i</sub>''' of the vector lattice '''L''' is called the '''point lattice'''. | Given an arbitrary point ''P'' in point space, the set of all the points ''P<sub>i</sub>'' deduced from one of them by a translation '''PP<sub>i</sub>''' = '''t<sub>i</sub>''' of the vector lattice '''L''' is called the '''point lattice'''. | ||
| Line 16: | Line 16: | ||
If the basis is primitive, the unit cell is called the ''[[primitive cell]]''. It contains only one lattice point. If the basis is non-primitive, the unit cell is a ''[[centred lattices|multiple cell]]'' and it contains more than one lattice point. The multiplicity of the cell is given by the ratio of its volume to the volume of a primitive cell. | If the basis is primitive, the unit cell is called the ''[[primitive cell]]''. It contains only one lattice point. If the basis is non-primitive, the unit cell is a ''[[centred lattices|multiple cell]]'' and it contains more than one lattice point. The multiplicity of the cell is given by the ratio of its volume to the volume of a primitive cell. | ||
| − | |||
| − | |||
| − | |||
== See also == | == See also == | ||
*[[Polar lattice]] | *[[Polar lattice]] | ||
*[[Reciprocal lattice]] | *[[Reciprocal lattice]] | ||
| − | |||
| − | |||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Latest revision as of 13:42, 10 November 2017
Réseau direct (Fr). Direktes Gitter (Ge). Reticolo diretto (It). 直格子 (Ja). Red directa (Sp).
Definition
The direct lattice represents the triple periodicity of the ideal infinite perfect periodic structure that can be associated to the structure of a finite real crystal. To express this periodicity one calls crystal pattern an object in point space En (direct space) that is invariant with respect to three linearly independent translations, t1, t2 and t3. One distinguishes two kinds of lattices, the vector lattices and the point lattices.
Any translation t = ui ti (ui arbitrary integers) is also a translation of the pattern and the infinite set of all translation vectors of a crystal pattern is the vector lattice L of this crystal pattern.
Given an arbitrary point P in point space, the set of all the points Pi deduced from one of them by a translation PPi = ti of the vector lattice L is called the point lattice.
A basis a, b, c of the vector space Vn is a crystallographic basis of the vector lattice L if every integral linear combination t = u a + v b + w c is a lattice vector of L. It is called a primitive basis if every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c. Referred to any crystallographic basis the coefficients of each lattice vector are either integral or rational, while in the case of a primitive basis they are integral. Non-primitive bases are used conventionally to describe centred lattices.
The parallelepiped built on the basis vectors is the unit cell. Its volume is given by the triple scalar product, V = (a, b, c).
If the basis is primitive, the unit cell is called the primitive cell. It contains only one lattice point. If the basis is non-primitive, the unit cell is a multiple cell and it contains more than one lattice point. The multiplicity of the cell is given by the ratio of its volume to the volume of a primitive cell.